La corde dont la présence en Loge fait l'unanimité des Francs-maçons, n'en demeure pas moins un symbole méconnu à bien des égards. Elle est cependant, pour de nombreuses traditions, le principe organisateur du ciel et de ses constellations. Outil symbolique de construction de l'initié, la corde à noeuds ouvre le chemin de la connaissance des métiers et celui de la magie par la confection de noeuds : éléments qui allient à la fois les concepts les plus avancés de la science moderne aux perceptions les plus anciennes de la tradition. La corde aux lacs d'amour par ses deux formes, sur le sommet des murs du temple et autour du tableau de Loge, nous questionne : d'où viennent-elles ? Quelles sont leurs fonctions rituelles ? N'y aurait-il pas une loi de la corde à respecter pour découvrir l'amour vrai ?
Ouvrage remarquable mettant en lumière la portée symbolique et rituelle de ce grand classique de la tradition spirituelle, ainsi que son indéniable sagesse.
Placée à l'Orient du temple, en regard de la Pierre brute, La Pierre cubique pourrait tout simplement être considérée comme l'une de ses émanations. Suivre cette voie, n'est pas rester dans une vision allégorique et se priver ainsi d'un véritable trésor de la pensée symbolique.
Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary.
Key Features:
- The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings
- Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra
- Explicit formulas are extended to apply to the geometric, spectral, and dynamic zeta functions associated with a fractal
- Examples of such formulas include Prime Orbit Theorem with error term for self-similar flows, and a tube formula
- The method of diophantine approximation is used to study self-similar strings and flows
- Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions
Throughout new results are examined. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts.
The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics.
This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions.
The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.
